Braking action of a brake disc is a local contacting problem, which brings influence to the modes and inherent frequencies of the system. The method of local discontinuous basis function can be used to estimate the modes of system accurately. In this paper, a brake disc, contacted by a brake boot or friction block, is simplified as a one-dimensional beam model with loop structure. The equation of motion is established. Firstly, the modes of the free vibration beam with no contact, defined as reference system, are calculated. Then the contact of the friction block is simplified as several linear springs. Considering the continuous conditions, the discontinuous modes are calculated and orthogonalized with the reference modes.,as the supplement of reference mode, used to calculate the system response. Compared with the finite difference method, Local discontinuous basis function method is more accurate than the traditional modal method.Analyses show that the local contact by the friction piece can restrain the symmetry of vibration of the brake plate, which leads to disappearance of sine or cosine modes. Based on the discontinuous mode analysis, vibration of the brake disk under action of external excitation is calculated. And the function of the nonlinear stiffness and friction, can make the vibration to be wave type.Further researches on structure unstable vibration mechanism of the brake noise can be carried out based on the above result.